On the fundamental group of a mapping space. An example

C ^OMPOSITIO M ^ATHEMATICA

V AGN L UNDSGAARD H ANSEN On the fundamental group of a mapping space. An example

Compositio Mathematica, tome 28, n

^o

1 (1974), p. 33-36

<http://www.numdam.org/item?id=CM_1974__28_1_33_0>

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33

ON THE FUNDAMENTAL GROUP OF A MAPPING SPACE.

AN EXAMPLE

Vagn Lundsgaard

^Hansen

Noordhoff International Publishing Printed in the Netherlands

1. Introduction

The purpose of this paper is ^to

provide

^an

example

^{of a}

mapping

space between

connected, compact polyhedra having finitely generated homotopy

groups in all

dimensions,

^{such that at least one}

component

in the

mapping

space has

infinitely generated

fundamental _group.

Throughout

^{K is a}

connected, compact polyhedron. Mostly

^{K =}

Si,

the unit circle. _{For any}

connected, topological

space

X, XK

shall denote the _{space of} continuous _{maps of K} into X

equipped

^{with the}

compact-

open

topology.

^In

particular XS1

denotes the free

loop

space ^on X. We

adopt

^the

terminology

^{that an}

aspherical

space is ^a

connected, compact

polyhedron X,

^{which is also an}

Eilenberg-MacLane

space

of type (03C0, 1),

i.e.

03C01(X) =

^{x and}

7ri(X)

^{= 0 for all i}

^~

^2.

THEOREM: There exists an

aspherical

space X with

finitely generated fundamental

group, such that

the free loop

space

XS1

contains a component

with

infinitely generated fundamental

group.

A similar

phenomenon

^cannot

happen

ⁱⁿ

higher

dimensions. It

is,

in

fact,

easy to prove that if X has

finitely generated homotopy

groups in

dimensions ~ 2,

^{then all the}

homotopy

groups of

XK

in

dimensions ~

²

are also

finitely generated.

Results on the

qualitative

^{structure of the}

homotopy

groups of ^a

mapping

space ^were obtained among others

by

^Federer

[1 J

^{and Thom}

[4].

^The

example

^{in the} theorem above shows that Thom’s statement

([4],

^Theorem

4)

is incorrect for the fundamental _group.

Finally,

^{I want to thank J.}

Eells,

^{A. C.}

Robinson,

^B.

Hartly

^and

D. B. A.

Epstein

for various

helpful

^remarks

during

^the

preparation

^of

this paper.

2. Free

loop

spaces. Their fundamental groups

In this section X is a

connected, topological

space with base

point

^*.

It is well-known that the

(path-)components

^{in the}

^mapping

^space

^Xs’

can be enumerated

by xi (X, *).

The fundamental _{group of} a

component

in

XS1 depends normally

^{on the}

component

ⁱⁿ

question.

^To

^describe how,

34

recall that the centralizer of an element _g in a group G is the

subgroup

^of

G defined

by Cg(G) = ^{{g’ ~ Glg’g}

⁼

^gg’l.

^{We have then}

PROPOSITION 1:

Suppose

^that

1t2(X, *)

^{= 0 and let}

^{f : S1 ~}

^{X be an}

arbitrary

^base

point preserving

map

representing

^the

homotopy

^class

[f] E 7r,(X, *).

^Then

PROOF: Let 03A9X denote the

ordinary loop

space ^on

X,

^{i.e. the} space of based _{maps of}

Si

into X

equipped

^{with the}

compact-open topology.

^{It is}

well-known that the map p :

XS1 ~

X defined

by

evaluation at the base

point

^of

S’,

also denoted _*, is ^a Hurewicz fibration with fibre 03A9X. See e.g. Hu

([2],

^{Theorem 13.1 p.}

83).

^{Consider now the}

^following part

^of

the

homotopy

^{sequence of that}

^fibration,

Since 03C01(03A9X, f) ~ 03C02(X, *)

⁼

^{0, p*} is a monomorphism. Hence 03C01(XS1,

f)

^is

isomorphic

^{to the}

image of p*.

^{To determine this}

image

observe that

a e xi

(X, *)

^{is in the}

image of p*

^{if and}

only

^if there exists a

map S1 ^{x sl}

~ X such that

S1 {*} ~

^X

represents

^a

and {*}

^x

^{S1 ~}

^X

^represents

[f].

^{On the} other hand such a map exists if and

only

^if the Whitehead

product

^{of a and}

[f] is

^the

identity

^element

of 03C01(X, *).

^{Since a Whitehead}

product

^{of elements in a} fundamental _group coincides with the cor-

responding

commutator

product,

^we

conclude,

^{that a is in the}

image

^of

p* if and only if 03B1[f]03B1-1[f]-1 =

^{1 or}

equivalently

^{that a E}

C[f] (03C01 (X, *)).

This proves the

proposition.

3. The

example

Let X be the

connected, compact,

2-dimensional

polyhedron

^obtained

from a

cylinder

^{over an} oriented circle with base

point by pinching

^one

boundary

^{circle into a}

figure eight

^{and then}

identifying

^the

resulting

^three

boundary

^circles

according

^to orientation. See

Figure

^1.

Fig. 1

We shall show that this

polyhedron

satisfies the

requirements

^{in the}

theorem. This will follow

easily

^from

Proposition

^{1 and}

Propositions

²

and 3 below.

The base

point

^{from the circle in the}

cylinder

leaves X with a base

point.

Denote the fundamental group of X

by

^03C0, and consider the elements x

and y

^{in x}

generated by

^{the two}

loops

^{indicated in}

Figure

^{1. For each}

n ⁼

0, 1, 2, ...

^we

put yn

⁼

x"yx-".

^In

particular

yo = y.

PROPOSITION 2: ^x is

generated by

^{the elements x and} y

subject

^{to the}

single

^relation

x-1yx

⁼

y2.

The centralizer

of

y ^E 03C0,

Cy(03C0),

^is

infinitely generated

with yo, y,, yn, ··· ^{as a} system

of

generators.

PROOF: X can be obtained from the

wedge

^{of the two}

loops generating

x

and y by adding

^{on a}

single

^2-cell

according

^{to a} map with the

homotopy

class

x- lyxy-2.

^{Hence the statement about x is}

well-known,

^see e.g.

Spa-

nier

([3],

^Theorem

^10,

p.

147).

x is an element of infinite order in ^x. Let F be the infinite

cyclic

^sub-

group of ^03C0

generated by x

^{and let C be the}

subgroup

^{of 03C0}

generated by

the elements y0, y1, ···, yn, - - -.

Clearly

^{F n} C contains

only

^{the neu-}

tral element. One proves

easily

^{that any word in x can be}

expressed

ⁱⁿ

the form

xmw,

where m is an

integer

^{and w is a} word in C. Hence 7r is

generated by

^{F and C.}

Altogether

^{03C0 is therefore a} semidirect

product

^of

the

subgroups

^{F and C.}

Using

^{the relation}

x-1yx

⁼

y2

it is easy

to prove that y2n+1

⁼

yn, and

hence that _yn ⁼

(yn+k)2k’ ^Having

^this information one verifies that C is abelian.

Since non-trivial powers of x ^{never commute}

with y

⁼

yo, it

^{is now}

clear that

Cy(03C0)

^{= C.}

Clearly

^{C is}

infinitely generated,

^{since one can never}

produce

^{the ele-}

ment yn+ 1 ^out of a

system

^{of éléments in C}

involving only

yo , ’ ’ ’, yn . This proves

Proposition

^2.

PROPOSITION 3: X is an

Eilenberg-MacLane

^space

of

^type

(03C0, 1).

PROOF: Let X’ denote the

polyhedron

obtained from the space in

Figure

¹

by identifying

^{this time}

only

^{the two}

right-hand

^{circles. Con-}

sider then the double infinite

telescope

^{X" obtained}

by glueing together copies

of X’ in both

ends, always

^{such that the} left-hand circle in _{any copy} of X’ is identified with the

pair

of identified

right-hand

^{circles in the}

neigh- bouring

copy of X’ ^to the left in the

telescope.

Pushing

^one

stage

^{from the left to the}

right

^{in the}

telescope

^{defines in}

the obvious _way an action of the

integers

^{Z on} X". The

quotient

space for this action can

clearly

be identified with X.

Furthermore,

^the

quotient

36

map X" ~ X is a

covering projection.

To prove that X is an

Eilenberg-

MacLane space

of type (03C0, 1),

it suffices therefore to prove that

03C0n(X") =

0 for

all n ~

^2.

For that _purpose, let

f :

^{Sn ~ X" be an}

arbitrary

^{based map of the}

n-sphere

^Sn

for n ~

^{2 into X".}

By

^a

compactness argument

^the

image

^of

this map lies in a finite

stage

^{of the}

telescope.

^{Therefore it} is easy ^{to see} that we can

homotope f into

^a map g : Sn ~

X",

^which maps ^one hemi-

sphere

^{of ,Sn into a}

long string,

and maps the other

hemisphere

^{of S" into}

a circle

separating

^two

copies

^{of X’}

sufficiently

^{far to the}

right

^{in the}

telescope. Furthermore,

^{we can} arrange

that g

maps the common boun-

dary

^{of the two}

hemispheres

^{into a}

single point.

^{It is} then clear

that g,

and hence

also f,

^is

homotopic

^{to the constant map. Therefore}

nn(X")

^{= 0}

for

all n ~ 2,

^{and as}

already

^{remarked this}

completes

^the

proof

^{of Pro-}

position

^3.

We are now

ready

^{for the}

PROOF OF THE THEoREm:

By Propositions

^{2 and}

3,

^{X is an}

aspherical

space with

finitely generated

fundamental _group.

Let 1;, : ^{S1 ~}

^{X be a based map}

representing

^the

homotopy

^class

[fy] =

^YEn

= 03C01(X,*). By Proposition 1, 03C01(XS1, fy)=Cy(03C0),

which is in-

finitely generated by Proposition

^{2. Hence the}

component

^of

XS1,

^which

contains the

map fy,

^has

infinitely generated

fundamental _group. Therefore X satisfies the

requirements

ⁱⁿ the theorem.

REFERENCES

[1] ^H. FEDERER: A study of function _spaces by spectral sequences. Trans. Amer.

Math. Soc. 82 (1956) ^340-361.

[2] ^{S. T. Hu:} Homotopy Theory. ^Academic Press, ^1959.

[3] E. H. SPANIER: Algebraic Topology. McGraw-Hill, ^1966.

[4] ^{R. THOM:} L’homologie ^des espaces fonctionnels. Colloque ^de topologie algé- brique, ^{Louvain 1956.} Thone, Liège; Masson, Paris, 1957, pp. 29-39.

(Oblatum: 4-X-1972 &#x26; 1-VIII-1973) Matematisk Institut K0benhavns Universitet K0benhavn

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